Theories and models in cognitive psychology tend to be nonlinear, e.g. Process Dissociation Model of Memory, but statistical methodology for assessing these theories is based on linear models. As a consequence, psychologists using nonlinear models have no suitable means of accounting for extraneous variability from the selection of items and participants. In these nonlinear contexts, unaccounted variability often leads to asymptotic bias in estimation and may lead to flawed inference in hypothesis testing. To fill this void, we propose a series of hierarchical nonlinear models to capture psychological processes of interest. Although models are custom-tailored for specific applications, the form of these models and the corresponding analytic techniques will have broad applicability across experimental psychology. Our overall strategy is to place linear models on parameters in nonlinear processes. For example, to account for item and participant variability in a signal detection analysis of memory performance, we assume that each individual and each item have separate effects on sensitivity. These items and participant effects are assumed to be random effects from parent distributions and modeled accordingly. The overall benefit is accurate estimation and vastly improved inference. In the course of specifying these models for psychological process, we necessarily make significant progress in Bayesian methodology, most notably in improving mixing and implementing Bayes Factor computations in hierarchical models with non-informative priors.